Conceiving Mathematical Terms and Propositions in the 14th Century

What is it about?

A metalinguistic shift occurred in 14th-century natural philosophy. Predominantly manifesting within a nominalist setting, it entailed a greater focus on propositions and terms referring to things or events in nature. This semantic turn extended to one of the foremost debates in natural philosophy: the continuum controversy. Thinkers aligning with nominalist inclinations wondered, for instance, if points are indivisible elements of a line and answered this question by focusing on the term ‘point’, i.e., on the issue of what ‘point’ signifies and stands for. Assuming a parsimonious ontology with no room for geometric items such as ‘point’, the problem was how to grant verifiability to mathematical statements housing empty mathematical terms. My article delves into the three main semantic strategies addressing this difficulty. While all views assume geometric items to lack genuine existence, they disagree on the way propositions featuring mathematical terms should be considered. A first strategy regards mathematical propositions as false, turning them into conditionals. The second one conceives of mathematical propositions as categorical relying on the imaginability of the clause, its terms, and its referents. The last one considers mathematical propositions as embedding terms stripped of their own supposition yet connoting a referent by indicating the lack of some (mathematical) feature in it. I will retrace the positions adopted by key historical actors that engaged in this debate, e.g., William of Ockham, John Buridan, Nicole Oresme. Concurrently, I will show how my overview is consistent with the perspectives articulated in the early 15th-century commentary on Aristotle’s Physics by John Marsilius.

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